 Pendulum approximation of a mean-motion resonance

Keeping one resonant harmonic and treating orbital eccentricity as fixed reduces mean-motion resonance dynamics to a physical pendulum equation. For the inner first-order term $\mathcal{R}=(G{M}_2/a_2)f(\alpha )e\cos \varphi$, leading Lagrange planetary equations give $\ddot{\varphi }\simeq 3j^2{\mu }_2{n}^2\alpha fe\sin \varphi$. Around its stable phase, the small-amplitude libration frequency is $jn\sqrt{3{\mu }_2\alpha |f|e}$. Evolution of eccentricity, other harmonics, and approach to a separatrix change this approximation.